Hyperreflexivity of the derivation space of some group algebras

Alaminos, J.; Extremera, J.; Villena, A. R.

doi:10.1007/s00209-009-0586-8

Cited by 11 publications

(19 citation statements)

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“…The following lemma is proven in [3,Lemma 5.1] for the case where X = A. Our proof is the generalization of their proof and is presented here for the sake of completeness.…”

Section: It Is Clear That Dist R (T S) ≤ Dist(t S)mentioning

confidence: 88%

“…The author would like to thank Armando R. Villena for providing the preprints of the manuscripts [1], [2], and [3]. The author also would like to thank the referee for carefully reading this article and providing helpful comments.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…On the other hand, it was shown in [11] that every local derivation on B(X), for a Banach space X, is a derivation. For further results see [3], [7], [14], [15], and reference therein.…”

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confidence: 98%

“…The main idea used there was to introduce the concept of bounded approximately local derivations and show that they have to be derivations. In [3,Corollary 5.3], J. Alaminos, J. Extremera, and A. R. Villena showed that in the case where…”

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confidence: 99%

“…We first recall and/or generalize what we require from [3] and make some careful estimates. Then we again use the existence of locally unital bounded approximate identities in group algebras of groups with polynomial growth and show that Z 1 (L 1 (G), X) is hyperreflexive if G is an amenable group with an open subgroup of polynomial growth and X is the dual of an essential L 1 (G)-bimodule.…”

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confidence: 99%

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Reflexivity and hyperreflexivity of bounded $N$-cocycles from group algebras

Samei

2011

Proc. Amer. Math. Soc.

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Abstract. We introduce the concept of reflexivity for bounded n-linear maps and investigate the reflexivity of Z n (L 1 (G), X), the space of bounded ncocycles from L 1 (G) (n) into X, where L 1 (G) is the group algebra of a locally compact group G and X is a Banach L 1 (G)-bimodule. We show that Z n (L 1 (G), X) is reflexive for a large class of groups including groups with polynomial growth, IN-groups, maximally almost periodic groups, and totally disconnected groups. If, in addition, G is amenable and X is the dual of an essential Banach L 1 (G)-bimodule, then we show that Z 1 (L 1 (G), X) satisfies a stronger property, namely hyperreflexivity. This, in particular, implies thatThe concept of reflexivity for linear subspaces of bounded operators on Banach spaces has its origin in operator theory. Let X be a Banach space, and let A ⊂ B(X) be an algebra of bounded operators on X. Let Lat A denote the set of the closed subspaces of X invariant under A; i.e. for every T ∈ A and I ∈ Lat A we have T (I) ⊂ I. We say that A is reflexive if A is the algebra generated by Lat A; i.e. every bounded operator satisfying T (I) ⊂ I for every I ∈ Lat A belongs to A. This concept is closely related to the well-known invariant subspaces problem: whether a bounded operator T ∈ B(X) has an invariant subspace.In [10], D. R. Larson generalized the concept of reflexivity, both algebraically and topologically, to subspaces of B(X, Y ) for Banach spaces X and Y . One motivation was to study the local behavior of derivations from a Banach algebra A to a Banach A-bimodule X. The main question that one asks is, for which algebras every so-called "local derivation" is a derivation, or equivalently, which algebras have algebraically reflexive derivation spaces? One can also ask the topological version of this question; i.e. when is the linear space of bounded derivations reflexive [10]?In the last two decades, the question of (algebraic) reflexivity of the derivation space has received considerable attention from various researchers, and some very interesting results have been obtained. In [9], R. D. Kadison showed that bounded local derivations from a von Neumann algebra into any of its dual bimodules are derivations. Kadison's result was generalized later by showing that the space of bounded derivations from a C * -algebra into any of its modules is both algebraically

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“…The following lemma is proven in [3,Lemma 5.1] for the case where X = A. Our proof is the generalization of their proof and is presented here for the sake of completeness.…”

Section: It Is Clear That Dist R (T S) ≤ Dist(t S)mentioning

confidence: 88%

Section: Acknowledgmentsmentioning

confidence: 99%

“…On the other hand, it was shown in [11] that every local derivation on B(X), for a Banach space X, is a derivation. For further results see [3], [7], [14], [15], and reference therein.…”

mentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

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